Levine, HI; Petters, AO; Wambsganss, J, *Applications of singularity theory to gravitational lensing. I. Multiple lens planes*,
Journal of Mathematical Physics, vol. 34 no. 10
(1993),
pp. 4781-4808 .
**Abstract:**

*The basic local and global features of stable multiple plane gravitational lens systems are investigated using tools from singularity theory. All stable multiple plane time-delay and lensing maps are classified, and the following global facts are proven under the weaker assumption of local stability. First, every locally stable multiple plane lensing map has an even number of cusps whether the associated deflector is singular or not. Second, for nonsingular deflectors the sum of the projectivized rotation numbers of its caustics is zero, while for singular ones it is negative and even. Third, if the deflector has g point masses on a single plane, then g is given by the formula g=-1/2Σcr(c), where r(c) is the projectivized rotation number of the critical curve c and the sum runs through all critical curves. Fourth, explicit counting formulas and bounds are found for the number of cusps for certain caustic networks. Finally, the latter yields that two point masses on a single lens plane will generate at least six cusps. However, if the masses are put genetically on separate lens planes, then there are at least eight cusps. © 1993 American Institute of Physics.*